A204155 Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-j, 2j-i), as in A204154.

1, -1, -7, -3, 1, 33, 39, 6, -1, -135, -255, -125, -10, 1, 513, 1323, 1092, 305, 15, -1, -1863, -6075, -7047, -3444, -630, -21, 1, 6561, 25839, 38610, 27135, 8946, 1162, 28, -1, -22599, -104247, -190593, -175230

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• tabl
• sign
• look

Comments

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

Examples

			Top of the array:
     1,   -1;
    -7,   -3,    1;
    33,   39,    6,   -1;
  -135, -255, -125,  -10,    1;
		

References

• (For references regarding interlacing roots, see A202605.)

Links

• Robert Israel, Table of n, a(n) for n = 1..10010 (rows 1 to 140, flattened)

Crossrefs

Cf. A204154, A202605, A204016.

Programs

Maple

f:= proc(n) local P,lambda,i;
P:= (-1)^n*LinearAlgebra:-CharacteristicPolynomial(Matrix(n,n,(i,j) -> max(2*i-j,2*j-i)),lambda);
seq(coeff(P,lambda,i),i=0..n);
end proc:
map(f, [$1..20]); # Robert Israel, Dec 03 2017

Mathematica

f[i_, j_] := Max[2 i - j, 2 j - i];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204154 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                 (* A204155 *)
TableForm[Table[c[n], {n, 1, 10}]]